Normality-preserving endomorphism-invariant subgroup: Difference between revisions

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==Definition==
A [[subgroup]] <math>H</math> of a [[group]] <math>G</math> is termed a '''normality-preserving endomorphism-invariant subgroup''' if, for every [[normality-preserving endomorphism]] <math>\sigma</math> of <math>G</math>, <math>\sigma(H)</math> is contained in <math>H</math>. A normality-preserving endomorphism is an endomorphism with the property that the image of any [[normal subgroup]] is normal.
==Examples==
===Extreme examples===
* The trivial subgroup is normality-preserving endomorphism-invariant in any group.
* Every group is normality-preserving endomorphism-invariant in itself.


{{subgroup property}}
===Examples arising from stronger properties or subgroup-defining functions===


==Definition==
* All [[fully invariant subgroup]]s, including the [[derived subgroup]] (commutator subgroup), as well as members of the [[derived series]] and [[lower central series]], are normality-preserving endomorphism-invariant.
* The [[Fitting subgroup]] and [[solvable radical]] are both normality-preserving endomorphism-invariant. In fact, they are something stronger: [[weakly normal-homomorph-containing subgroup]]s.


A [[subgroup]] <math>H</math> of a [[group]] <math>G</math> is termed a '''normality-preserving endomorphism-invariant subgroup''' if, for every [[normality-preserving endomorphism]] <math>\sigma</math> of <math>G</math>, <math>\sigma(H)</math> is contained in <math>H</math>.
===Examples in small finite groups===


==Formalisms==
{{subgroup property see examples embed|normality-preserving endomorphism-invariant subgroup}}


{{frexp}}
{{subgroup property}}


{| class="sortable" border="1"
! Function restriction expression !! <math>H</math> is a normality-preserving endomorphism-invariant subgroup of <math>G</math> if ... !! This means that the property is ... !! Comments
|-
| {{invariance-short|normality-preserving endomorphism}}
|-
| {{endo-invariance-short|normality-preserving endomorphism}}
|}
==Relation with other properties==
==Relation with other properties==


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|-
| [[Weaker than::Weakly normal-homomorph-containing subgroup]] || image of subgroup under a homomorphism that sends normal subgroups inside it to normal subgroups is normal || [[weakly normal-homomorph-containing implies normality-preserving endomorphism-invariant]] || [[normality-preserving endomorphism-invariant not implies weakly normal-homomorph-containing]] || {{intermediate notions short|normality-preserving endomorphism-invariant subgroup|weakly normal-homomorph-containing subgroup}}
| [[Weaker than::Weakly normal-homomorph-containing subgroup]] || image of subgroup under a homomorphism that sends normal subgroups inside it to normal subgroups is normal || [[weakly normal-homomorph-containing implies normality-preserving endomorphism-invariant]] || [[normality-preserving endomorphism-invariant not implies weakly normal-homomorph-containing]] || {{intermediate notions short|normality-preserving endomorphism-invariant subgroup|weakly normal-homomorph-containing subgroup}}
|-
| [[Weaker than::Normality-preserving endomorphism-balanced subgroup]] || any normality-preserving endomorphism of the whole group restricts to a normality-preserving endomorphism of the subgroup || || || {{intermediate notions short|normality-preserving endomorphism-invariant subgroup|normality-preserving endomorphism-balanced subgroup}}
|}
|}


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! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
|-
|-
| [[Stronger than::Strictly characteristic subgroup]] || invariant under all [[surjective endomorphism]]s || [[normality-preserving endomorphism-invariant implies strictly characteristic]] || [[strictly characteristic not implies normality-preserving endomorphism-invariant]] || {{intermediate notions short|strictly characteristic subgroup|normality-preserving endomorphism-invariant subgroup}}
| [[Stronger than::strictly characteristic subgroup]] || invariant under all [[surjective endomorphism]]s || [[normality-preserving endomorphism-invariant implies strictly characteristic]] || [[strictly characteristic not implies normality-preserving endomorphism-invariant]] || {{intermediate notions short|strictly characteristic subgroup|normality-preserving endomorphism-invariant subgroup}}
|-
| [[Stronger than::direct projection-invariant subgroup]] || invariant under all projections to [[direct factor]]s || [[normality-preserving endomorphism-invariant implies direct projection-invariant]] || [[direct projection-invariant not implies normality-preserving endomorphism-invariant]] || {{intermediate notions short|direct projection-invariant subgroup|normality-preserving endomorphism-invariant subgroup}}
|-
| [[Stronger than::finite direct power-closed characteristic subgroup]] || in any finite [[direct power]] of the whole group, the corresponding power of the subgroup is characteristic || [[normality-preserving endomorphism-invariant implies finite direct power-closed characteristic]] || [[finite direct power-closed characteristic not implies normality-preserving endomorphism-invariant]] || {{intermediate notions short|finite direct power-closed characteristic subgroup|normality-preserving endomorphism-invariant subgroup}}
|-
|-
| [[Stronger than::Characteristic subgroup]] || invariant under all [[automorphism]]s || (via strictly characteristic) || (via strictly characteristic) || {{intermediate notions short|characteristic subgroup|normality-preserving endomorphism-invariant subgroup}}
| [[Stronger than::characteristic subgroup]] || invariant under all [[automorphism]]s || (via strictly characteristic) || (via strictly characteristic) || {{intermediate notions short|characteristic subgroup|normality-preserving endomorphism-invariant subgroup}}
|-
|-
| [[Stronger than::Normal subgroup]] || invariant under all [[inner automorphism]]s || (via characteristic) || (via characteristic) || {{intermediate notions short|normal subgroup|normality-preserving endomorphism-invariant subgroup}}
| [[Stronger than::normal subgroup]] || invariant under all [[inner automorphism]]s || (via characteristic) || (via characteristic) || {{intermediate notions short|normal subgroup|normality-preserving endomorphism-invariant subgroup}}
|}
|}

Latest revision as of 14:32, 8 July 2011

BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]

Definition

A subgroup H of a group G is termed a normality-preserving endomorphism-invariant subgroup if, for every normality-preserving endomorphism σ of G, σ(H) is contained in H. A normality-preserving endomorphism is an endomorphism with the property that the image of any normal subgroup is normal.

Examples

Extreme examples

  • The trivial subgroup is normality-preserving endomorphism-invariant in any group.
  • Every group is normality-preserving endomorphism-invariant in itself.

Examples arising from stronger properties or subgroup-defining functions

Examples in small finite groups

Below are some examples of a proper nontrivial subgroup that satisfy the property normality-preserving endomorphism-invariant subgroup.


Below are some examples of a proper nontrivial subgroup that does not satisfy the property normality-preserving endomorphism-invariant subgroup.


This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Fully invariant subgroup invariant under all endomorphisms fully invariant implies normality-preserving endomorphism-invariant normality-preserving endomorphism-invariant not implies fully invariant |FULL LIST, MORE INFO
Normal-homomorph-containing subgroup any homomorphic image of the subgroup that's normal in the whole group is contained in the subgroup normal-homomorph-containing implies normality-preserving endomorphism-invariant normality-preserving endomorphism-invariant not implies normal-homomorph-containing |FULL LIST, MORE INFO
Normal-subhomomorph-containing subgroup any homomorphic image of a subgroup that's normal in the whole group is contained in the subgroup (via normal-homomorph-containing) (via normal-homomorph-containing) |FULL LIST, MORE INFO
Weakly normal-homomorph-containing subgroup image of subgroup under a homomorphism that sends normal subgroups inside it to normal subgroups is normal weakly normal-homomorph-containing implies normality-preserving endomorphism-invariant normality-preserving endomorphism-invariant not implies weakly normal-homomorph-containing |FULL LIST, MORE INFO
Normality-preserving endomorphism-balanced subgroup any normality-preserving endomorphism of the whole group restricts to a normality-preserving endomorphism of the subgroup |FULL LIST, MORE INFO

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
strictly characteristic subgroup invariant under all surjective endomorphisms normality-preserving endomorphism-invariant implies strictly characteristic strictly characteristic not implies normality-preserving endomorphism-invariant |FULL LIST, MORE INFO
direct projection-invariant subgroup invariant under all projections to direct factors normality-preserving endomorphism-invariant implies direct projection-invariant direct projection-invariant not implies normality-preserving endomorphism-invariant |FULL LIST, MORE INFO
finite direct power-closed characteristic subgroup in any finite direct power of the whole group, the corresponding power of the subgroup is characteristic normality-preserving endomorphism-invariant implies finite direct power-closed characteristic finite direct power-closed characteristic not implies normality-preserving endomorphism-invariant |FULL LIST, MORE INFO
characteristic subgroup invariant under all automorphisms (via strictly characteristic) (via strictly characteristic) |FULL LIST, MORE INFO
normal subgroup invariant under all inner automorphisms (via characteristic) (via characteristic) |FULL LIST, MORE INFO
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